Uniform impedance network



June 18, 1940.

. H. A. WHEEL-En UNIFORM IMPEDANCE NETWORK Filed Aug. 26, 1937 2 Sheets-Sheet 1 FIG.5.

INVENTOR HARLD A. wH

ATTORNEY June 18, 1940.

H. A. WHEELER UNIFORM IMPEDANCE NETWORK 2 Sheets-Sheet 2 Filed Aug. 26, 195'? INVENTOR HA OLD A. WHE "ER ATTORNEY Patented June -18, 1940 UNITED STATES PATENT OFFICE 2,204,712 UNIFORM IMPEDANCE NE'rwoRrt ware Application August 26,

12 Claims.

This invention relates generally to uniform impedance networks and is particularly useful in matching the image impedance of an exponential line with the impedance of a-circuit or with the image impedance of another line to which the exponential line is to be connected, particularly a uniform line having a purely resistive and uniform image impedance. The invention is particularly suitable for utilizing an exponential transmission line as means for coupling an antenna to a uniform line going to the terminal circuit of a modulated-carrier signal apparatus.

In order that two transmission circuits may be coupled together in such manner that there are no reflections from the junction, it is necessary that the circuits have, at their junction, image impedances which are properly matched over the entire range of frequencies to be transmitted. Exact matching over a wide range of frequencies is impossible if the image impedances of the circuits to be connected have dissimilar variation with frequency. For example, it is impossible if only oneof the image impedances has a complex value. If theimage impedances are both purely 25 resistive over the range of frequencies for which image impedance matching is required, the circuits can be matched by the use of filter sections, including a transformer if it is necessary to change the impedance level at the junction. However, a transformer can beincluded only in a band-pass filter of limited frequency range and for this reason may be unsatisfactory where it is required to pass a wide range of frequencies.

It has previously been proposed to use a transmission line having distributed impedance which varies exponentially with the length of the line for coupling two circuits which have different impedance levels. One such arrangement is disclosed in United States Patent No. 1,927,522, granted September 19, 1933, on an application of Nils E. Lindenblad. It is there proposed to utilize a simple impedance-matching filter or line device to match the image impedance of an exponential line with the impedance of one of the circuits to be coupled. However, it is now recognized that an exponential line of generalized length has an image impedance which is not a pure resistance and which cannot, therefore, be matched with the impedance of a load circuit by a simple filter.

Various attempts have been made to obtain approximate matching between such a line and the circuit to which it is coupled. In United States Patent No. 2,018,320, granted October 22, 1935, on' an application of Walter van B. Roberts,

1937, Serial No. 161,017

it is suggested that the image impedance may be kept nearly pure resistance if the rate of taper is made sufiiciently small and the line is made sufiiciently long to secure the desired impedance ratio.

An approximate matching of impedance between an antenna and a radio receiver by means of a tapered line is disclosed in Radio Service Notes of General Electric Company (1935), at pages 13, 14, and 15, in an article entitled V- Doublet' Antenna System. However, this line comprises tapered straight conductors and, thus, is only a poor approximation of the coupling which would be provided by an exponentially ta-. pered line.

It has also been proposed to use an exponential line connecting an antenna with the input circult of a modulated-carrier signal receiver, but in each of the previous proposals the line is not ideally matched at its junctions with the adjacent circuits. Such a proposal is found in the Roberts patent mentioned above. As another example, it hasbeen proposed to provide an antenna coupling circuit wherein a single wire having an exponentially tapered impedance to ground is used, in which case the capacitance and inductance exist with respect to ground. Such a disclosure is found in Australian Patent No. 18,994, accepted August 30, 1929. However, in these cases, the exponential line does not have at its junctions even approximate matching of image impedance over a considerable range of frequency.

It is an object of the present invention, therefore, to provide a uniform impedance network having a substantially uniform responsiveness over a wide range of frequencies.

It is another object of the present invention, therefore, to provide a terminating circuit foran exponential transmission line of any taper and length, by means of which the matching of image impedances is independent of frequency.

It is a further object of the invention to provide an antenna-coupling circuit, the transmission band of which covers a wide range of frequencies.

In accordance with the invention, means are provided for terminating an exponential line of any taper and length, by means of which the image impedance of the line can be exactly matched by a filter having either a mid-shunt or a mid-series termination. In a transmission line having such a termination, ordinary filter theory with reference to confluent constant-k and mderived filters is applicable.

pedance over the range of frequencies to be I transmitted can be exactly matched with an exponential line of any length and of such taper as to pass the given range of frequencies. Since the impedance level is modified in the exponential line without the undesirable filter characteristics resulting from the use of a transformer for changing the impedance level, an exponential line having terminal circuits in accordance with the invention provides an improved means for connecting two circuits having lmped'ances of different levels and provides an ideal means of connecting two circuits having resistive image impedance terminations of different levels; and provides an ideal means of connecting two circuits having resistive image impedance terminations of different impedance levels.

7 Further, in accordance with the invention, means are provided whereby an exponential transmission line terminates in an image impedance almost exactly 'matching that of a doublet antenna for all frequencies of the'transmission band. v e

For a better understanding of the present invention, together with other and further objects thereof, reference is had to the following description, taken in connection with the accompanying drawings, and its scope will be pointed out in the appended claims.

Fig. 1 of the'drawings is a schematic representation used for deriving the equations of an exponential line; Fig. 2 is a schematic diagram of the image impedance of an exponential line resolved into series components; Fig. 3 is a schematic diagram of the image impedances of an exponential lineresolved into parallel components; Fig. 4 is a graph of the image impedances of an exponential transmission line both below and .'above the cutoff frequency; Fig. 5 illustrates a circ'uit arrangement for matching the line image impedance with the corresponding constant-k filters by resolving into series components; Fig. 6 illustrates a circuit arrangement for obtaining an approximately constant image resistance; Fig. '7 shows a balanced disposition of the elements of Fig. 6; Fig. 8 is a graph of the relation between wire separation and distance from the. point of maximum separation for lines having various f given ratios of maximum separation to wire diame ter; Fig. 9 illustrates as a special case one of the transmission lines, the characteristics of which are shown in Fig. 8; Fig. 10 shows a cir cuit arrangement for substantially,matching the impedance of an exponential line with a doublet antenna; and Fig. 11 shows a circuit arrangement for matching a vertical-wire antenna with a single-wire exponential line having a ground return circuit.

In setting forth the invention, it is necessary I is no dissipation and that the line is tapered in accordance with the formulaez in which L1 and C1 are the inductance and capacitance per unit of length, 2: is the distance along the wires from the narrow end (the end denoted by subscript a), and )\c is the cutoff wavelength to be identified below. It is also assumed that the separation is much less than the wave length in space at all frequencies of interest.

The usual line equations are:

in which E and I are vectors of alternating voltage and current along the line. There is assumed, subject to justificationbelow, a progressive wave, free of terminal reflections, starting at the narrow end (subscript a) E: Zflz/m-fiz/X) a 2w(-z/)\ jZ/ v in which A is the wave length along the line at the operating frequency. Also,

Substituting in Equations 2, the exponential terms cancel out, thereby justifying the assumed progressive wave of Equations 3. There remain the relations,

in which we is the cutoff frequency of the line and v is the steady-state wave velocity along the wires, having a limiting value of vo at high fre quencies. Therefore, from (5) and (6), the limiting velocity (at infinite frequency) is "o /w n I and the actual velocity is v=v /1 w c 7 7 notes attenuation as distinguished from wave transmission.

In the case mentioned, the steady-state velocity is variable and is greater than the limiting velocity which is nearly the velocity of light. 'It is a form of distortion which is absent in the ideal uniform line,.but is found in filters. The

group velocity or impulse velocity does not really exceed that of light, although the phase velocity or "steady-state velocity are greater, as effected by the taper.

The image impedance of the line at the narrow end is the driving-point impedance of the till) square root term is imaginary and negative.

progressive wave assumed in (3); it is obtained by solving (4) and substituting from (6):

In further expressions, it is convenient to use the image resistance R]: of any small section of the line, so short that its taper can be neglected:

The line is comparable with a transformer whose turns ratio is Tzw/mz arl n 2.7310. (11) in which I is the length of the line. It appears that a nominal ratio can be secured with a length only a small fraction of An.

In the transmission band, above the cutofi frequency we, the exponental line has a complex im;

row) and high-impedance (wide) ends, is given by the following expressions, in which the subscripts a and 1) denote the respective ends:

The latter is obtained in the same manner as the former, the opposite sign of we denoting the efiec- -tive opposite taper toward the backward wave.

1 mission band above the cutoif frequency, the

square root term is real and positive. In the attenuation band below the cutoff frequency, the The remaining term in the above expressions is imaginary and its sign. depends on which end of the These relations are valid only in the transmission band, where Ba and Rh are equal to the midshunt elements.

The real comseries image resistance of the corresponding filters.

Fig. 3 shows the resolution of Zn and Zb into parallel components, as follows:

These relations are valid only in the transmission band, where Re and Re are equal to the mid shunt image resistance of. the corresponding filter.

In Figs. 2 and 3, the quantities Ca, La and Ch, Lb are the full-series and full-shunt elements of the corresponding filter. These quantities doubled are the respective mid-series and mid- Thisrelation forms the basis for matching the corresponding line and filter terminations.

Fig. 4 shows graphically the image impedance and its components at either end of the exponential line. The abscissae are inversely proportional to frequency, to show conciselythe limiting conditions at high frequency. The abscissae between and 1 thus represent the transmission band of the exponential line. It willbe seen that in this band (the image impedance of either end of the line is a constant, the image impedances being represented by-Zs' and Zb, respectively. In the transmission band the image impedance Za can be resolved into series components Ra'and Xa, or parallel components Ra and- X8195 illustrated. Similarly, the impedance Za. in the transmission band can be. resolved into either series or parallel components, as shown. In the attenuation band, that' is, beyond point I on the abscissae, the image impedance of the line at either end is a pure reactance and is shown represented by Za or X9. and Zb or Xb for the respective ends of the line.

Matching the exponential line with adjacent circuits involves not only the relative magnitudes of the impedances at the junction, but also the nature of the line impedance. The line impedance approximately matches a constant resistance (Rka or Rkb) only at frequenciesmuch higher than the cutofi frequency. Closer matching is secured by power-factor correction which converts the line impedance to a pure resistance variable with frequency and by means of which either end of the line can be matched exactly with a high-pass constant-k filter. An m-de rived termination of such a filter can be used to match closely a constant resistance.

Fig. shows the principle of power-factor correction of the line impedance. The image impedance at the narrow end of the line is represented, as in Fig. 2, by Ra in series with 20a (in dotted lines). The effect of reactance of 2G,; is cancelled by connecting 2L1 in parallel, while the resistance Re. is transformed to Re, which is the mid-shunt image resistance of the corresponding constantJc filter, by means of the half-section high-pass filter 3 .formed by 2C, and ZLa. The wide end is also represented, as in Fig. 2, by Rb in series with -2Cb. The effect of the reactance -2Cb is cancelled by connecting 201. in series, leaving purely resistive image impedance Rh, which is the mid-series image resistance of the corresponding constant-1c filter.

image impedance of which is Ra. At the wide end,

Rmis the mid-series image impedance of the halfsection'i, the mid-shunt image impedance of which is Rb. Any number of half-sections may be added, according to the ordinary rules of filter designi The two essential requirements of impedance matching are met by the line and filters of Fig. 5. First, each terminal circuit presents to the line an image impedance equal to that of a continuation of the line with the same exponential taper. For example, at the wide end, 20:; and Rh in series result in the same image impedance at the narrow end as a continuation of the line at the wide end without interruption. Secondly, each end of theline with its power-factor correction presents to the respective filter an image impedance equal to that of the adjoining filter termination.

Since the constant-k image Ra, Ra, Rb, and Rb vary with frequency, they are not very nearly matched with a constant-resistance circuit. The matching is greatly improved by the use of an m-type half-section betweenthe constant-resistance circuit and the constant-k image resistance of the line or filter section. In Fig. 5, for example, the half-section at either end maybe replaced by an m-type half-section. The complete circuit at either end then becomes that of Fig. 6. The circuits of Fig. 7 are equivalent to those of Fig. 6, but are arranged to have bilateral symmetry so that the system may be balanced relative to ground. Figs. 6 and '7 provide nearly exact matching, at frequencies the ends of the exponential line and the constant resistances Rka and Rkb- The choice of the filter design parameter m determines the residual relative variations of the image resistances at the terminations in Figs. 6 and '7. A value of m ,equal to 0.6 is suggested, which causes the image resistances to be within about 5 percent of the respective terminating resistances Rka or Rkb at all frequencies more than 15 percent above the cutoff frequency. This value of m also places the frequency of infinite attenuation 20- percent below the cutoff frequency.

The circuits of Figs. 6 and '7 are about the simplest that can be used to match closely either end of an exponential line with a constant-resistance circuit, such as a uniform line. If an exponential line without the matching expedients of Fig. 5 or Fig. 6 is connected directly to a constant-resistance circuit, the matchingwill not be satisfactory at frequencies less than 40% above the cutoff frequency because the power factor of the line impedance is then substantially less than '70 percent. If the circuit arrangement of Fig. 5 is used to match an exponential line with a constantresistance circuit, so that the effective image impedance of the exponential line is purely resistive but variable. with frequency, the constant-resistance circuit should have a resistance value which is the geometric mean of the minimum and maximum values of the nonuniform effective image resistance of the exponential line, over the required frequency range. For exampie, 1f the system is to pass all frequencies more resistances above the cutoff frequency, between than i40% ,above the cutoff frequency, the constantre'sistance should be about 0.84 Rka to match Ra. or about 1.19 Rka. to match Re.

A two-wire transmission line has an exponential taper only when the separation of the wires varies correctly with distance along the wires. If the desired variation of Rk is given, in accordance with Equation 2, the actual values of R1; deterniine'fthe ratio of separation to Wire diameter at all distances. An explicit formula for the separation in terms of the distance, may be derived on the assumption of bare wires of zero resistance.

Referrii'ig to Fig. l and Equation 2,

in which a is the wire diameter, y is half the separation, and z is the distance measured along the wire. This equation yields the solution,

There is a maximum separation, beyond which this formula cannot be satisfied, because the separation would have to increase more rapidly than the distance along the wire, The maximum separation is defined by the relation:

d v i=1 25 From Equation 24 dy rd 4 4y. @=-Tc10g (7)4120: (26) Since .2 is measured from an arbitrary datum point, Equations 24 and 26 can be rewritten as:

where z=0 at the point of maximum separation. This greatly simplifies Equation 26 and enables its solution simultaneously with Equation 24. This solution involves the wire diameter d, the cutoff wave length kc, and the maximum halfseparation 11111. It is expressible in any of several forms, but an explicit solution for ym is impossible.

The separation of the wires along the length of the exponential line is best expressed in terms of the ratio y/ym, which is always less than unity. In order to make this expression as simple as possible, )2 is measured from the point of maximum separation. At all points of less separation and, therefore, at all other possible points,

g as a parameter.

, 4 meters.

explicit solution for y.

arcs) in which I. Um 4....

loglT The following formula gives an explicit solution for z. 1

I log. i =log. %405 logl The shape of an exponential line near the point of maximum separation (z'=0), as computed by Formula 34, is nearly independent of the wire diameter d, in practical cases where the latter is very small as compared with the maximum halfseparation ym. A convenient criterion for the shape is the ratio yin/d. .Fig. 9 shows the shape of an exponential line having a ratio ym/d=1000, which is a practical value.

Fig. 8 is plotted from Equation 36 in such a manner as to show thevariation of separation with distance along the wires from the point of maximum separation. The ratio ym/d is used Interpolation is possible for values of this ratio between 100 and 10,000.

These curves are most useful in the design. of exponential lines.

Fig. 10 illustrates the matching of a uniform transmission line ID with a straight-wire doublet H, 12. The following procedure is useful in designing an exponential line l3 for this use, ac-

cording to the above principles and relations. The cutoff wave length \=100 meters, the wire diameter (1:0.2 centimeter, and the uniform line separation D=10 centimeters are assumed. "The ratio xc/100d is 500. By plotting Equation 0,1/m/d is found to be 520; from Equation 31, 7\c/ym is found to be.96; therefore, the half maximum separation ym is equal to 104 centimeters.

The minimum half-separation is D/2 =5 centimeters=.048 ym. Reading opposite y/!1m=.048 in Fig. 8, it is found that the exponential taper will require a length of wire equal to 3.9 1. m, about Also, the variation of separation along the wires is determined from the curves of Fig.

iisconnected in this space.

is required at the doublet end of the tapered r line of Fig. 10, because the rate of taper decreases gradually in the doublet itself. The above design 8. The shape is substantially like that shown in Fig. 9.

Tension cords or separators ll are connected between the tapered wires, at intervals sufficiently close to maintain substantially the correct shape. The straight wire of the doublet is cut out in the center at l5 for a space of about 2 meters, and the wide end of the tapered line 13 No matching filter ground return and having its constants exponentially varying along its length.

While there have been described what are at present considered the preferred embodiments of this invention, it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from this invention, and, therefore, it is aimed in the appended claims to cover all such changes and modifications as fall within the true spirit and scope of this invention.

What is claimed is:

1. A wave-signal translating system comprising a transmission line having an impedance which varies exponentially along the length of the line and a terminating circuit for one end of said line comprising auxiliary reactance means proportioned to adjust the image impedance at said end to an impedance exactly equal, over all frequencies of 1thetransmission band of saidline, to the image impedance of a constant=k high,- pass filter section having the same cutoii frequency.

2. A wave-signal translating systemcomprising a transmission line having an impedance which varies exponentially alongthc length of the line and a terminating circuit for said line comprising an inductance having a value of i e la in parallel with said line at the end of the line having the lower impedance level, where we is the cutoff frequency of said line and Lia and Cm are. respectively, the inductance and capacitanceper unit length of said line at said end, saidfirst mentioned l e lh in series with said line at the end of the line having the higher impedance level, where'we is the cutoff frequency of said line and Lip and Cw are.

. respectively, the inductance and capacitance per unit length of said line at said end. said firstmentioned capacitance thereby adjusting the image impedance of said line at said, end to a pure resistance over the transmission band.

4. A wave-signal translating system comprising a transmission line having an image impedance which varies exponentially along the length of the line and a terminating circuit for said line comprising an inductancehaving a value of per unit length of said line at said end, said firstmentioned inductance thereby adjusting the impedance of said line at said end to match exactly over the transmission band the mid-shunt image impedance ofa constant-k filter having a seriesreactance arm' with a capacitance of 'I I 5. A'wave-signal translating system compris ng a transmission line having an impedance which varies exponentially along the length of the line anda' terminating circuit for said line comprising a' capacitance having a value of i e ib in ar swith as line at the end of the line having' the higher impedance level, where we is the cutoff frequency of said line and Lib and] C11) are, respectively, the inductance and capacitance per unit length of said line at said end, said firstmentioned capacitance thereby adjusting the 'gje'impedance of said line at said end to match lyfover the transmission band the mid-series "series-reactance arm with a capacitance of i a e ib and a shunt-reactance arm with an inductance of l I 2 .2 e Clb 6: Anelectrical impedance network comprisiliary filter network coupled .ing a terminal circuit including reactance tending to limit the responsiveness of said terminal circuit over awide band of frequencies, an auxto said terminal circuit and including said reactance as a fullterminating reactance atone end of said filter,

. said filter having an m-derived termination at its other end, and a resistive-terminating circuit coupledft'o said other end of said filter and U matching the impedance thereof over the pass hand of said filter, whereby the'responsiveness of said network is maintained substantially uniform over said pass band.

7. An electrical impedance network comprising a terminal circuit including reactance ef- -fectivelyin shunt across said terminal circuit tending to limit the'responsiveness of said ter- 'minal 'c i rcuit over a wide band'of frequencies,

an auxiliary filter network coupled to said ter-; minal ci'rcuit'and including said reactance as afull-shunt terminating element at one end of said-filter, said filter having an m-derived termination at its other end, and a'terrninating resistor coupled to said other end of said filter and matching the impedance thereof over the pass band of said filter, whereby the responsiveness of-said network is maintained substantially uniform ,oversaid pass band.

8. An electrical impedance network comprising a terminal circuitincluding reactance effectively in series in saidterminal circuit tending to limit the responsiveness of said terminal circuit 7 over a wideband of frequencieaan auxiliary filter network coupled to :said terminal circuit and including said reactance as a full-series terminating element at one end of said-filter, said filter having an m-derived termination at its other end,- and a terminating resistor coupled to said otherend of said filter and matching the impedance .thereof over the pass band of said filter, whereby the responsiveness of said netna eimpedance of a constant-k filter having a r work is' maintained substantially uniform over saidipass band.

9. An electrical impedance network comprising .a terminal circuit including an inductive ele mentefi'ectively coupled across said terminal circuit and tending to limit the responsiveness of said terminal circuit over a wide band of frequencies, an auxiliary filter network coupled to said terminal circuit and including said inductive element as a full-shunt terminating element at one end of said filter, said filter having an m-derived termination at its other end, and a terminating resistor coupled to said other end of said filter and matching the impedance thereof over the pass band of said filter, whereby the responsiveness of said network is maintained substantially uniform over said pass band.

10. A signal-translating system for operation over a wide range of frequencies comprising a pair of terminals between which there is effectively a substantial reactance tending to limit the response of said system over said range, a filter having a predetermined image impedance over said range and coupled to said pair of terminals, said filter comprising a portion of said reactance as a mid-element of said filter, and an impedance termination coupled to the dead end of said filter proportioned substantially to match the image impedance of said filter over said range, the reactive constants of said dead-end filter being so proportioned relative to said reactance and the operating frequency range that the mean value of the impedance between said pair of terminals over said range isyapproximately the limiting value that can be maintained therebetween over said range.

11. A signal-translating system for operation over a wide range of frequencies comprising a pair of terminals in series with which there is efiectively substantial capacitance tending to limit the response of said system oversaid range, a filter having a predetermined image impedance over said range coupled to said pair of terminals, said filter comprising a portion of said capacitance as a mid-series element of said filter, and an impedance, termination coupled to the dead end of said filter proportioned substantially to match the image impedance of said filter over said range, the reactive constants of said dead-end filter being so proportioned relative to said capacitance and the operating frequency range that the mean value of the admittance between said pair of terminals over said range is approximately the maximum that can be maintained therebetween over said range.

12. A signal-translating system /for operation over a wide range of frequencies comprising a pair of terminals across which there is effectively substantial inductance tending to limit the response of said system over said range, a filter having a; predetermined image impedance over said range coupled to said pair of terminals, said filter comprising a portion of said inductance as a mid-shunt element of said filter, and an' impedance termination coupled tothe deadend of said filter proportioned substantially to match the image impedance of said filtenover said range, the reactive constants of said 'deadend filter being so proportioned relative to said inductance and the operating frequency range that the mean value of the impedance between said pair of terminals'over saidrange is approxr Certificate of Correction Patent No. 2,294,912. June 18, 1940.

HAROLD A. WHEELER It is hereby certified that errors appear in the printed specification of the above numbered patent requiring correction as follows: Page 2, first column, line 45, for

the word impedance read impedances; page 4, second column, lines 25 and 37, Equations 24 and 26 respectively, for

g read zfy lines 42 and 45, Equations 27 and 28 respectively, for

read and that the said Letters Patent should be read with these corrections therein that the same may conform to the record of the case in the Patent Office.

Signed and sealed this 3rd day of September, A. D. 1940.

[SEAL] HENRY VAN ARSDALE,

Acting Commissioner of Patents. 

